So far we have considered a band-limited channel that does not introduce any signal distortions in its passband. Now we will generalize our considerations to the channel band-limited to B Hz, which has a given transfer functionH (f ). Assume that the input signal is Gaussian, it has a power density spectrum P (f ) and its power is limited to P. This means that,
BP (f )df =P. The signal is disturbed by additive Gaussian noise characterized by the power density spectrumGn(f ). An example of such a characteristic is shown in Figure 1.27. In order to determine the channel capacity we divide the channel band into N frequency intervals of width f so that B =N f. If N is sufficiently high, the width of a single component channel is so small that it has approximately flat characteristics and we can use (1.111) to derive its capacity. Thus, transmission through the channel with the transfer function H (f ) can be treated as a parallel transmission through N ideal passband channels of bandwidth f. Using formula (1.111) we can obtain the following formula for the capacity of theith component channel
Ci = flog
"
1+ f P (fi)|H (fi)|2 f Gn(fi)
#
(1.116) If the channel input signal is characterized by the power density spectrum P (f ) then the power density spectrum at the output of the channel with the transfer functionH (f )
H(f)
∆f f
B
Figure 1.27 Example of the channel transfer function of the channel with bandwidthB Hz
is given by the expressionP (f )|H (f )|2. Therefore, the power of the signal seen at the output of the ith component channel of bandwidth f with the center frequency fi is f P (fi)|H (fi)|2. The capacity of the whole channel approximated byN ideal passband channels is equal to
C= N
i=1
Ci = f N
i=1
log
"
1+P (fi)|H (fi)|2 Gn(fi)
#
(1.117) If the bandwidth f of component channels tends to an infinitely low value df, then the sum evolves into the integral and discrete frequency valuesfi change into a continuous variable f. Finally, we obtain the following formula for the channel capacity
C=
$
B
log
"
1+P (f )|H (f )|2 Gn(f )
#
df (1.118)
The capacity of the channel with transfer function H (f ) depends both on its char- acteristics and the power density spectra of the input signal and noise. The properties of physical channel and noise are often difficult to change; however, it is possible to change the power density spectrum P (f ) of the input signal. Recall that the capacity calculations require the input signal to be Gaussian. Thus, we would like to determine the power density spectrum of the input signal for which the channel capacity is maximum, i.e. the highest number of bits in a time unit that can be transmitted over the channel H (f ). Searching for the best shape of the power density spectrumP (f ) that maximizes capacity (1.118) with the assumption that the signal power is constant and is equal to P is an optimization problem with a constraint. The solution method is similar to the method that was applied in derivation of the probability density function for which the differential entropy is maximized.
Let us apply Theorem 1.12.1 again. This time we deal with maximization of function of form
C=
$
B
F (f, P (f ))df =
$
B
log
"
1+P (f )|H (f )|2 Gn(f )
#
df (1.119)
for the constraint
$
B
ϕ(f, P (f ))df =
$
B
P (f )df =P (1.120)
As we remember from Theorem 1.12.1, we find the bestP (f )by solving equation (1.93).
In our case this equation has the form
∂
∂P (f ) -
log2
"
1+P (f )|H (f )|2 Gn(f )
#.
+λ ∂
∂P (f )
P (f )
=0 (1.121)
In equation (1.121) functionP (f )is treated as a variable. The calculation of the derivative with respect toP (f )leads to the following equation
log2eã Gn(f )
Gn(f )+P (f )|H (f )|2ã|H (f )|2
Gn(f ) +λ=0 (1.122) Substituting 1/K = −λ/log2ewe receive
|H (f )|2
Gn(f )+P (f )|H (f )|2 = 1
K (1.123)
which after simple calculations leads to the formula P (f )=K− Gn(f )
|H (f )|2 (1.124) If the additive noise is white, i.e.Gn(f )=N0/2, then
P (f )=K− N0/2
|H (f )|2 (1.125)
Substituting (1.124) into the equation describing the constraint,
BP (f )df =P, we end up with
$
B
"
K− Gn(f )
|H (f )|2
#
df =P (1.126)
From this formula the following expression arises (see Figure 1.28) KB=P+$
B
Gn(f )
|H (f )|2df (1.127) Let us note that KB is the area of a rectangle of width B, which is limited by the horizontal axis and the horizontal straight line located at the height K. The second term of (1.127) is the area under the curveGn(f )/|H (f )|2. Thus, the input signal powerP is the area denoted in grey color above the mentioned curve, which fills out the area above the curve to the levelK (Figure 1.28).
The analysis of the optimized input signal power density shape, which leads to the maximum capacity of the channel with a given characteristic H (f ) and noise power density Gn(f ), leads us to interesting conclusions. It turns out that the channel capacity is maximized if we assign the highest input signal power to the channel sub-bands with the lowest attenuation of the input signal. Less power should be placed in those frequency intervals in which the signal is heavily attenuated. It is against our intuition, because at first glance it seems that we apparently should amplify the transmitted signal in those frequency ranges in which the channel attenuates it heavier. The process of shaping of the input signal power density is calledpower loading. The rule of power loading reminds
P(fk)
fk f
PowerP K
B Gn(f)
|H(f)|2
Figure 1.28 Illustration of the choice of input signal power density maximizing the capacity of the channel with a given transfer function
us of pouring water into a basin, therefore this rule is often known as thewater pouring principle. Power loading is performed in the frequency domain. However, we will learn in the next section that it is also possible in the time domain.
Let us note that power loading requires a feedback channel from the receiver back to the transmitter. In order to assign the input signal power optimally, the receiver has to derive the channel characteristic (we say that it performs channel estimation) and then it has to transmit it back to the transmitter via a feedback channel. Thus, it is a case in which Channel State Information (CSI) is known both to the transmitter and receiver. In suboptimum systems the channel state information is known only at the receiver and it can be applied only in the signal detection. In this case power loading is not possible and the feedback channel is not required.